Rotational Motion Notes

Introduction to Rotational Motion

• Review of linear motion concepts:
  • Displacement: Change in position of an object.
  • Velocity: Speed and direction of travel.
  • Forces: Causes of accelerated motion.
  • Inertia and momentum: Inertia in motion.
• These concepts are revisited in the rotational plane with changes to names but a similarity in concept.

Rotational vs. Revolving

• Rotation: Movement around an internal axis (e.g., a bicycle tire rotating around its axis).
• Revolution: Movement around an external axis (e.g., a red spot on the wheel revolving about the axis).

Earth’s Rotational Motion

• The Earth revolves around the sun every $365 \frac{1}{4}$ days.
• The Earth rotates around its axis every 24 hours.

Period (T)

• Definition: Time for one revolution (or the time to go around once).
• Units: Seconds, minutes, hours, or any unit of time.

Frequency

• Definition: How often an object travels around a circle.
• Units: Revolutions per minute (rpm) or revolutions per second (rps).
• Relationship between Frequency and Period:

Period vs. Frequency - Example

• If it takes 5 seconds for a dog to go around a merry-go-round once (the Period), we can determine the frequency.

The Radian

• Definition: The angle for which the length of a circular arc is equal to the radius of the circle.
• Units for measuring angles: Degree (°), revolution (rev), and radian (rad).
• Radian is the most convenient unit for angle measurements in scientific calculations.
• Arc Length Formula: $S = r\theta$

Describing Angular Motion

• Angles are indicated around the circumference of the circle in both degrees and radians.
• Radians are the preferred unit for Physics.
• Circumference Formula: $C = 2\pi r$
• Conversion: 1 revolution = 360° = $2\pi$ radians

Rotational vs. Tangential Velocity

• Tangential velocity (v): Describes the motion of an object along the edge of a circle; direction is always along the tangent to that point.
  • V is dependent on the point's location relative to the axis of rotation.
• Rotational velocity ($\omega$): Describes the motion of a rotating body.
  • The entire body rotates at the same $\omega$.

Rotational Velocity

• Definition: The number of rotations per unit of time.
• Formula: $\omega = \frac{2\pi}{T}$
• Units: radians per second (rad/sec)
• Also referred to as angular velocity.

Tangential Speed

• Definition: The speed of an object moving along a circular path.
• Formula: $v = \frac{2\pi r}{T} = \omega r$
• Units: m/s (meters per second)

Rotational Inertia

• An object rotating about an axis tends to remain rotating about the same axis at the same rotational speed unless interfered with by an external influence.
• Definition: The property of an object to resist changes in its rotational state of motion (symbol I).
• Also known as the Moment of Inertia.

Factors Affecting Rotational Inertia

• Mass of the object.
• Distribution of mass around the axis of rotation.
• The greater the distance between an object’s mass concentration and the axis, the greater the rotational inertia.

Implications of Rotational Inertia

• The greater the rotational inertia, the harder it is to change its rotational state.
• Example: A tightrope walker carries a long pole with high rotational inertia for stability.

Rotational Inertia and Axis of Rotation

• Rotational inertia depends on the axis around which it rotates.
• Easier to rotate a pencil around an axis passing through it.
• Harder to rotate it around a vertical axis passing through the center.
• Hardest to rotate it around a vertical axis passing through the end.

Center of Gravity (CG)

• Definition: The average position of an object's weight distribution.
• For simple, uniform objects, the center of gravity is located at the geometric center.
• The center of gravity can be located outside of an object.

Locating the Center of Gravity

• An object hangs with the center of gravity below the point of suspension.
• An object will balance if pivoted exactly above or below its center of gravity.
• Balance on a pivot is stable if CG is below the pivot.

Human Center of Gravity

• Standing upright, your CG is roughly in the center of your body (at about 55% of your height).
• Location of your CG will shift when you bend your torso, move your arms and legs, etc.

Torque

• Definition: The tendency of a force to cause rotation.
• Depends upon three factors:
  • Magnitude of the force
  • The direction in which it acts
  • The point at which it is applied on the object

Torque Equation

• Formula: Torque = radius × force
• The radius depends upon where the force is applied and the direction in which it acts.
• The force needs to be applied perpendicular to the radius.
• Unit for torque is Newton-meter (N*m).

Lever Arm Examples

• Lever arm is less than the length of the handle because of the direction of force.
• Lever arm is equal to the length of the handle.
• Lever arm is longer than the length of the handle.

Centripetal Force

• Definition: Any force directed toward a fixed center.
• Centripetal means “center-seeking” or “toward the center.”
• Example: Whirling a tin can at the end of a string; you pull the string toward the center to keep the can moving in a circle.

Centripetal Force in Circular Motion

• For an object moving in a circle, there is an inward force acting upon it in order to cause its inward acceleration.
• For objects moving in circular motion, there is a net force acting towards the center which causes the object to seek the center.

Centripetal Force - Example

• When a car rounds a curve, the centripetal force prevents it from skidding off the road.
• If the road is wet, or if the car is going too fast, the centripetal force is insufficient to prevent skidding off the road.

Angular Momentum and Its Conservation

• Systems that can change their rotational inertia through internal forces will also change their rate of rotation.
• Formula: $L = \omega * I$

Angular Momentum - Example

• If, by pulling the weights inward, the rotational inertia of a man reduces to half its value, by what factor would his angular velocity change?

Rotational Motion Equations - CP